请教几道数分题
[*]设\(f'(x)\in C\)，求证：\(\D\int_0^1 x^n f(x)\dif x=\frac{f(1)}{n}+O\left(\frac1n\right)\).
[*]设\(f(x)\)在\(\)上单调，证明：\(\D\lim_{n\to\infty}\int_0^{2\pi}f(x)\sin(nx)\dif x = 0\).
[*]设\(f(x)\)在\(\)上连续，证明：\(\D\lim_{n\to\infty}\int_0^{2\pi}f(x)\sin(nx)\dif x = 0\).
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